On intertwining and w-hyponormal operators

• Autorzy: Otieno M.O.
• Języki publikacji: EN

Abstrakty

EN

Given A, B mem B(H), the algebra of operators on a Hilbert Spaces H, define deltaA,B : B(H) arr B(H) and DeltaA,B : B(H) arr B(H) by deltaA,B(X) = AX - XB and DeltaA,B(X) = AXB - X. In this note, our task is a twofold one. We show firstly that if A and B* are contractions with C.o completely non unitary parts such that X mem ker DeltaA,B, then X mem ker DeltaA*,B*. Secondly, it is shown that if A and B* are w-hyponormal operators such that X mem ker deltaA,B and Y mem ker deltaB,A, where X and Y are quasi-affinities, then A and B are unitarily equivalent normal operators. A w-hyponormal operator compactly quasi-similar to an isometry is unitary is also proved.

Słowa kluczowe

EN

w-hyponormal operators; contraction operators and quasi-similarity

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2005
• Tom: Vol. 25, no. 2
• Strony: 275--285
• Opis fizyczny: Bibliogr. 36 poz.

Twórcy

autor

Otieno M.O.

• University of Nairobi, Department of Mathematics, P.O. Box 30197, Nairobi, Kenya,

Bibliografia

• [1] Aluthge A.: On p-hyponormal operators for 0 < p < 1. Integral Equations and Operator Theory 13 (1990), 307-315.
• [2] Aluthge A.: Some generalized theorems on p-hyponormal operators. Integral Equ­ations and Operator Theory 24 (1996), 497-501.
• [3] Aluthge A., Wang D.: On w-hyponormal operators. Integral Equations and Ope­rator Theory 36 (2000), 1-10.
• [4] Aluthge A., Wang D.: On w-hyponormal operators II. Integral Equations and Operator Theory 37 (3) (2000), 324-331.
• [5] Aluthge A., Wang D.: An operator inequality which implies paranormality. Math. Inequality Appl. 2(1999), 113-119.
• [6] Ando T., Takahashi K.: On operators with unitary p-dilations Ann. Polon. Math. 66 (1997), 324-331.
• [7] Ando T.: On Hyponormal Operators. Proc. Amer. Math. Soc. 14 (1963), 290-291.
• [8] Berberian S.K.: Extensions of a theorem of Fug led e and Putnam. Proc. Amer. Math. Soc. 71 (1978), 113-114.
• [9] Berberian S.K.: Introduction to Hilbert Spaces. New York, Oxford Univ. Press 1961.
• [10] Clary S.: Equality of spectra of quasi-similar hyponormal operators. Proc. Amer. Math. Soc. 53 (1975), 88-90.
• [11] Douglas R. G.: On the operator S*XT = X and related topics. Acta Sci. Math. 30 (1969), 19-32.
• [12] Duggal B.P.: Quasi-similar p-hyponormal operators. Integral Equations and Operator Theory 26 (1996), 338-345.
• [13] Duggal B.P.: On Intertwining operators. Mh. Math. 106 (1988), 139-148.
• [14] Duggal B.P.: On generalised Putnam-Fuglede theorems. Mh. Math. 107 (1989), 309-332.
• [15] Duggal B. P.: A remark on generalised Putnam-Fuglede theorems. Proc. Amer. Math. Soc. 129 (2001), 83-87.
• [16] Duggal B. P.: On characterising contractions with C10 pure part. Integral Equ­ations and Operator Theory 27 (1997), 314-323.
• [17] Duggal B. P., Jeon I.H.: p-Hyponormal operators and quasi-similarity. Integral Equations and Operator Theory 49 (2004), 397-403.
• [18] Dunford D., Schwartz J.: Linear operators, part III: Spectral operators. New York, Interscience (1971).
• [19] Fan M.: An asymmetric Putnam-Fuglede theorem for 6-class operators and some related topics. J. Fudan Univ. Natur. Science 26 (1987), 347-350 (Chinese).
• [20] Fong C.K.: On M-hyponormal operator, Studia Math., 65 (1979), 1-5. [21] Halmos P. R.: A Hilbert Space Problem Book. Springer-Verlag, 1982.
• [22] Hoover T.B.: Quasi-similarity of operators. Illinois Jour. Math. 16 (1972), 678-686.
• [23] Chuan Hou J., On Putnam-Fuglede theorems for non-normal operators. Acta Math. Sinica 28 (1985), 333-340 (Chinese).
• [24] Jeon I. H.: Weyl's theorem and quasi-similarity. Integral Equations and Operator Theory 39 (2001), 214-221.
• [25] Jeon I.H., Tanahashi K., Uchiyama A.: On quasi-similarity for Log-hyponormal Operators. Glassgow Math. Jour. 46 (2004), 169-176.
• [26] Radjabalipour M.: Ranges of hyponormal operators. Illinois Jour. Math. 21 (1977), 70-75.
• [27] Radjabalipour M.: On majorization and normality of operators. Proc. Amer. Math. Soc. 62 (1977), 105-110.
• [28] Radjabalipour M.: An extension of Putnam-Fuglede theorem for hyponormal operators. Math. Z. 194 (1987), 117-120.
• [29] Radjavi H., Rosenthal P.: On roots of normal operators. J. Math. Anal. Appl. 34 (1971), 653-665.
• [30] Stampfli J.G., Wadhwa B.L.: An asymmetric Putnam-Fuglede theorem for do­minant operators. Indiana Univ. Math. Jour. 25 (1976), 359-365.
• [31] Tanahashi K.: On log-hyponormal operators. Integral Equations and Operator Theory 34 (1999), 364-372.
• [32] Uchiyama A., Tanahashi K.: Fuglede-Putnam theorem for p-hyponormal or log-hyponormal operators. Glassgow Math. Jour. 44 (2002), 397-410.
• [33] Wadhwa B.L.: M-hyponormal operators. Duke Math. Journal 41 (1974), 655-660.
• [34] Williams L.R.: Quasi-similarity and hyponormal operators. Jour. Oper. Theory 5 (1981), 678-686.
• [35] Xia D.: Spectral theory for hyponormal operators. Basel, Birkhauser (1981).
• [36] Yang L.: Quasi-similarity of hyponormal and subdecomposable operators Jour. Funct. Anal. 112 (1993), 204-217.